A tightly-wound solenoid of radius a and length l has n turns per unit length. It carries an electric current i . Consider a length dx of the solenoid at a distance x from one end, which contains n , dx turns and can be approximated as a circular current i , n , dx . Determine the total magnetic field at the centre of the solenoid by integrating the magnetic field due to this circular current under proper limits.

Step-by-step solution

Let the center of the solenoid be the origin. The distance of the elemental ring from the center is y . The magnetic field at the center due to an elemental ring of width dy at a distance y from the center is: dB = ₀ i (n , dy) a^2 2(a^2 + y^2)^ 3/2 Integrating from y = - l 2 to y = l 2 to find the total magnetic field: B = _ -l/2 ^ l/2 ₀ n i a^2 2(a^2 + y^2)^ 3/2 dy Let y = a , then dy = a ^2 , d . B = ₀ n i a^2 2 a ^2 (a^2 ^2 )^ 3/2 d = ₀ n i 2 , d B = ₀ n i 2 [ ] Substituting = y a^2 + y^2 and applying the limit